I relax by solving Rubik's cubes, and I wonder how many different ways there are to arrange a 3x3x3 Rubik's cube. There are 6 middle pieces. These are fixed in relation to each other, so there is only one possibility for the middle. There are 8 edges and 12 corners. These are different shapes so they can't be in each other's places. There are 8 different places for the first edge to be. There are then only 7 possible places to put the next one, as it can be placed in any of the other positions. There are 6 places to put the next, for the same reason, then 5, 4, 3, 2 and 1. That is 8 factorial (8!), which is 40,320. It is similar for the corners, as there are 12 places you can put the first corner, 11 for the second, and so on down to 1. 12! is 479001600. There are then 2 ways round you can put each of the 8 edges, so you multiply by 28, and 3 ways round you can put each corner, so there are 12! x 312 x 8! x 28 possible combinations, which equals 519,024,039,293,878,272,000. This is not actually the correct numbers of ways you can arrange a cube, because there are some positions you can't be in. You can't have just one edge flipped, just one corner flipped, or just two pieces swapped, so the actual sum is 12! x 311(8 x 7 x 6 x 5 x 4 x 3) x 27, which equals 43,252,003,274,489,856,000. That is the biggest number I've ever come across in maths problems!
